Jensen Alpha Calculator: Working the Residual Return

A Jensen alpha is the portfolio return less the return a beta of that size was expected to deliver. The expectation is the risk-free rate plus beta times the benchmark's excess over that rate. On the Anantara Multi-Asset Portfolio's stated year, 6.5 plus 1.08 times 6.1 gives 13.088 per cent, and 14.2 less 13.088 leaves plus 1.112 percentage points.

The defaults reproduce the whole worked instance in one screen. The Rs 500 crore mandate, of which Rs 300 crore sits in equity, returned 14.2 per cent for one stated twelve month period against a benchmark that returned 12.6 per cent. The gap is 1.6 points of headline outperformance. A risk-free rate of 6.5 per cent leaves the benchmark 6.1 points of excess, a beta of 1.08 scales that to 6.588, and adding the rate back gives an expected return of 13.088 per cent. The residual is 14.2 less 13.088, or plus 1.112 percentage points, of which the beta split puts about 0.49 points down to carrying more market exposure than the benchmark and about 1.11 points to the residual itself. Neither part is a statement about anybody's skill.

The meaning of the residual, and what a positive one does and does not establish about the person who produced it, is covered under alpha itself. The work here is what to do with four numbers, in what order, and what has to be printed alongside the answer for anybody else to check it.

Everything below runs on one record. The Anantara Multi-Asset Portfolio is an invented discretionary mandate of Rs 500 crore, run for a single institutional holder, an invented charitable endowment. For one stated twelve month period it returned 14.2 per cent. Its composite benchmark, 60 per cent a broad equity index and 40 per cent a broad bond index, neither of them named here, returned 12.6 per cent over the same twelve months. The risk-free rate for that period was 6.5 per cent. The betaThe slope from a regression of the portfolio's returns on the benchmark's returns. A slope of 1.08 says the portfolio moved 1.08 units for each unit the benchmark moved, on average, over the period the regression covered. of the portfolio against that benchmark was 1.08. The four figures belong together, and no one of them can be swapped for a figure from another year or another mandate.

The measure itself carries a name. The residual of a portfolio return against a market model expectation is Jensen alphaThe measure named for Michael C. Jensen, who set it out as the intercept left over once a market model has been fitted to a portfolio's returns., after Michael C. Jensen.

What are the four inputs, and what does each one do?

Four numbers go in. The portfolio return is what the mandate actually delivered over a stated window, and it is the only input that carries the result being explained. The risk-free rateThe return on the instrument the mandate treats as carrying no default risk, quoted for the same period as everything it is compared against. is what the same window paid for taking no market exposure at all. The rate appears twice in the working, a detail worth noticing at the outset. The benchmark return is what the composite delivered over that same window. And the beta is the estimated sensitivity of the portfolio to that same benchmark.

Three of those four are measurements, read straight off a valuation report, an index provider's series and the quote for the instrument the mandate names. The fourth is not a measurement at all. A beta is a regression slope, produced by fitting one return series against another over some window at some frequency, and a different window or a different frequency produces a different slope from the same underlying portfolio. A beta is an estimate wearing the clothes of a fact.

All four inputs must belong to one window, and the single most frequent way this calculation goes wrong is that it goes wrong before it starts, when a beta estimated over three years of monthly data is dropped into a working built on twelve months of returns. Nothing in the arithmetic will object. The subtraction still subtracts, the multiplication still multiplies, and a perfectly formatted number comes out at the end describing a portfolio that never existed: one with this year's returns and the last three years' market sensitivity.

Think of a shopkeeper working out whether this year was a good year. She takes this year's takings, and she compares them with what a stall of that size on that street ordinarily takes. If she uses this year's takings against a street footfall figure collected three years ago, before the flyover opened, her comparison is arithmetically flawless and completely empty. The window is not a technicality. The shared window is what makes the two numbers about the same world.

What is the order of operations?

Four steps, and they cannot be reordered without changing the answer. Step one: subtract the risk-free rate from the benchmark return to get the benchmark excessThe benchmark return less the risk-free rate over the same period. The excess is what the market paid over and above what taking no market exposure paid., which is 12.6 less 6.5, or 6.1. Step two: multiply that excess by the beta, giving 6.588. Step three: add the risk-free rate back on to reach an expected returnThe return the model says a portfolio of this sensitivity should have produced over this period. The expected return is constructed from the other three inputs, never observed. of 13.088 per cent. Step four: subtract that expectation from the portfolio return, leaving plus 1.112.

Step three is the one readers skip. Once the excess has been scaled, the risk-free rate has to go back in. Step one took it out. Left out, what has been computed is the expected excess rather than the expected return, and subtracting that from a full portfolio return compares two things that are not the same kind of number. The risk-free rate comes out at step one and goes back in at step three, and those two moves are the two appearances.

The answer is a difference between two percentages, so it is expressed in percentage pointsThe unit of a gap between two figures that are themselves percentages. A move from 12 per cent to 14 per cent is two percentage points, and calling it a two per cent rise would mean something different. and never as a percentage of anything. Plus 1.112 percentage points is the correct reading. Plus 1.112 per cent of the portfolio return would be a different quantity altogether. Write it that way and nobody can tell what base the percentage is taken on. The number stops being checkable.

Why does the beta scale the benchmark's excess and not the whole return?

The wrong version looks entirely reasonable. If the portfolio is 1.08 times as sensitive as the benchmark, why not multiply the benchmark return by 1.08 and be done?

Because a beta does not measure sensitivity to the benchmark's return. A beta measures sensitivity to the part of that return which is payment for carrying market risk, and that part is the benchmark's excess over the risk-free rate. The other part, the risk-free rate itself, is available to anybody who takes no market exposure at all. The risk-free rate is not something a beta is exposed to. A beta does not scale it.

Follow the consequence to its edge. The wrongness becomes obvious there without any figures. Multiplying the whole benchmark return by the beta would say that a portfolio with a beta of zero expects a return of zero, when a portfolio with no market exposure whatsoever plainly expects the risk-free rate. A pile of cash left in the instrument the mandate treats as free of default risk has a beta of essentially nothing and still earns 6.5 per cent over the stated year. Any working that hands it zero has been built wrong.

Run the wrong version deliberately, and label it as wrong. Multiply the whole benchmark return by the beta: 1.08 times 12.6 is 13.608. Take that off the portfolio return and the answer is plus 0.592, against the correct plus 1.112. One misplaced multiplication, 0.520 percentage points, identical inputs.

The gap has a shape worth carrying. The gap is exactly the risk-free rate multiplied by the beta less one: 6.5 times 0.08 is 0.52. Which means the error is zero when the beta is one, small when the beta is close to one, and larger the further the beta sits from one and the higher the risk-free rate stands. In an environment where the risk-free rate is near zero the mistake barely shows at all. Where it is 6.5 per cent, it moves the answer by nearly half of the answer.

How much does each input actually move the answer?

Take the inputs one at a time, hold the other three still, and watch what happens. The movement produced is the sensitivityHow far an output moves when one input is nudged and everything else is held where it was. Sensitivity is read one input at a time. Moving two at once settles nothing about either. of the output, and it is the part of a calculator that almost nobody prints.

The four rows are in the table below, and the calculator above names the move out loud with each edit to a field. The row worth pausing on is the risk-free rate, the only input that reaches the answer twice. Raising the rate by a point shrinks the benchmark excess by a point. The beta scales that into a 1.08 point fall in the expectation, and then the rate itself is added back a point higher. The two effects nearly cancel, and 0.08 points is all that survives.

Three sanity checks fall out of those rows, and together they make any alpha testable in the head. A point on the portfolio return moves alpha by a point. A point on the benchmark moves it by the beta. A point on the rate moves it by the beta less one. At a beta of one that movement is exactly zero.

The input that looks the most solid, the risk-free rate, barely touches the answer, while the input that looks the most technical, the beta, moves it a great deal. The care those two are usually given runs the other way round. Committee packs argue about which rate to use. The same packs very rarely ask over what window the beta was fitted. Yet swapping a 6.5 per cent rate for a 7.5 per cent one changes this answer by eight hundredths of a point. A beta estimate of 1.18 instead of 1.08 changes it by more than half a point.

How does the answer move as the beta changes?

Hold the three returns still and the whole working collapses into one line. Alpha is the portfolio's own excess over the risk-free rate, 14.2 less 6.5 or 7.7 points, minus the beta multiplied by the benchmark's excess of 6.1 points. So alpha equals 7.7 less 6.1 times the beta, and nothing else. Alpha against beta is a straight line sloping down, drawn below, and it is the single most useful relation on the record: at a beta of zero the whole 7.7 points would be residual, at the recorded 1.08 it is plus 1.112, and it reaches zero at 7.7 divided by 6.1.

The crossing point is computed rather than searched for: it is the ratio of the portfolio's excess over the risk-free rate to the benchmark's excess over the same rate, and there is never any need to hunt for it by trial. It also shows something the single answer does not. The recorded 1.08 sits 0.1823 below the crossing, so if the true sensitivity of this portfolio were 1.2623 the entire residual would vanish and the whole 1.6 points would be market exposure and nothing else.

For comparison, the record's own split of that 1.6 points using this beta puts about 0.49 points down to carrying more market exposure than the benchmark and about 1.11 points to the residual. The split is only as firm as the 1.08. The beta decomposition must never be mixed with the separate allocation and selection split of the same 1.6 points worked in attribution. Attribution answers a different question on a different base.

How is the working read backwards, from the answer to the inputs?

The four steps run one way, but the check runs the other way. The reverse check is the one used most often. Somebody else has usually done the working and handed over the result. The expected return and the residual must add back to the portfolio return exactly, with nothing left over. A sheet where they do not has an arithmetic fault in it rather than a debatable one. Here that is 13.088 plus 1.112, which is 14.2, and the calculator above prints that sum with every change to a field.

The same reversal answers a harder question. If a pack prints an alpha and a beta but never says what expected return it used, it can be recovered by subtracting the alpha from the portfolio return. Plus 1.112 taken off 14.2 gives 13.088. The expectation is the rate plus the beta times the benchmark excess, so from there any one remaining input can be worked out from the other two. The reversal is how a sheet is found to have used a benchmark return nobody mentioned, or a rate from a different quarter, without the formula it ran ever being shown.

What must be printed beside the answer?

An alpha of plus 1.11 percentage points, standing on its own on a slide, is not a computation. The figure is a claim. Nobody receiving it can reproduce it or tell whether the correct working or the shortcut produced it. The correct working and the shortcut gave plus 1.112 and plus 0.592 on identical inputs, and the bare number looks equally respectable either way.

An alpha printed alone cannot be checked by anybody. Being uncheckable is exactly what turns a computation into an assertion. The specification is eight items, not one. The four inputs. The window all four share. A description of the benchmark, unnamed on this record. And then the two that belong to the beta alone: the window it was estimated over and the frequency of the observations used.

The Anantara record carries six of the eight. Missing are the beta's estimation window and the frequency, and the honest thing is to print those two fields marked absent rather than leave them off the sheet. A missing field reads as a field that was satisfied. A field marked absent reads as what it is, a gap somebody should close before the number is relied on.

Who reads this number, and what do they do with it?

Rukmini Deshpande chairs the investment committee of the endowment that holds the Anantara Multi-Asset Portfolio. Faiz Ahmad Ansari runs the mandate and brings the pack. The alpha line arrives in a pack with a great deal else in it, and what she does with it is narrow and worth copying.

Her first move is not to read the number. She checks first that the four inputs printed under it share a window. Where they do not, the number below them is not about any period at all and the rest of the discussion is wasted. Her second move is to look for the beta's own window and frequency. Where those are absent, as they are here, the number is still usable, but it is usable as an estimate with an unstated tolerance rather than as a measurement, and she says so out loud so that the minute records it that way.

Her third move separates a committee that reads reports from one that is read to. She asks what alpha would be at a beta of 1.18 and at 0.98. The spread, plus 0.502 to plus 1.722, is the honest width of the answer, given how far a beta estimate moves with the window. A residual of plus 1.11 that could as easily be plus 0.50 supports a different conversation from one that is tight.

A credit officer does the same thing with a borrower and never calls it alpha. A workshop reports profits above what workshops of that size on that road usually make. Before anything is concluded, the officer checks that both figures cover the same twelve months, and asks how the road average was built, over what stretch and from how many workshops. If it came from a different year, or from four workshops, the excess is arithmetic performed on unrelated quantities. The defect is the same as a three year beta dropped into a one year working.

What can this calculator not establish?

The calculator cannot establish whether the beta it was handed is right. No test of the slope is available to it, and it will produce a clean, confidently formatted answer from a badly estimated one. A number displayed to three decimal places reads as though something has been measured to three decimal places. The third decimal here is downstream of a regression slope that would move in the second decimal if the estimation windowThe stretch of history a regression was fitted over, together with how often the observations were sampled. Change either and the slope that comes out changes too. changed by a few months.

The calculator cannot establish whether the residual repeats: one period produces one number, and that number says nothing about whether the same working next year returns plus 1.11 or minus 0.40. Nor can it establish what produced the residual. Security selection, a sector position, a timing decision, a stale valuation on an illiquid holding and a benchmark that never described the mandate properly all arrive at the same place in this arithmetic.

And it cannot establish whether the answer is good. Judging the answer needs an alternative to compare against, and this record does not contain one. The working can print the residualWhat is left of an outcome once a model has taken out everything it claims to account for. A residual is defined by the model rather than measured directly. together with how far it travels when each input is nudged.

A calculator that returns a single number with no width around it invites more confidence than its inputs can support. The honest output is the answer and the sensitivity together, never the answer alone. Plus 1.11 points, and: 0.61 points of movement for every 0.10 on the beta. The second line is not a caveat. The movement is half the result, and the calculator above prints it beside every edit.

The Anantara Multi-Asset Portfolio, the charitable endowment that holds it, its composite benchmark, Rukmini Deshpande and Faiz Ahmad Ansari are invented.
Educational material. Not advice on any investment, tax, budget or market position.